Let T be a metric space with distance function r(x,y) expressing the definitive predication that involves T with the real numbers, R. Therefore the juxtaposition of left and right hemispheres resonates in perfect accordance with the proposition that T and R are embedded simultaneously in the full structure of manifold M. Ergo we pass on to an enlargement *M of M, whereby the non-standard metric space is diffeomorphism invariant.

So if f(x) is a homeomorphism from T onto S, then for every point p in T, does f(u(p) = u(f(p)) ?

A metric space is a set of points such that for every pair of points, there is a nonnegative real number called their distance that is symmetric, and satisfies the triangle inequality, which states that the sum of the measures of any two sides of any triangle is greater than the measure of the third side. Space is then a tranformation[invariant]. Two objects with relative velocity will have a relative measure that transforms into the other. In effect, the separation does not exist in an extrinsic sense. ABC = BCA = CAB

So if f(x) is a homeomorphism from T onto S, then for every point p in T, does f(u(p) = u(f(p)) ?

Well, I don't know. You didn't tell us what "u" is.

Two objects with relative velocity will have a relative measure that transforms into the other. In effect, the separation does not exist in an extrinsic sense. ABC = BCA = CAB

"relative velocity"? How did "velocity" get into a discussion of metric spaces? In any case, I have no idea what you mean by "ABC = BCA= CAB" because you have not defined ABC, BCA, or CAB.

x^3 + y^3 = (x+y)*[(x+y)^2 - 3xy]

What are x and y? They are not points in the metric space you mentioned before because such operations are not defined in a general metric space. They certainly aren't numbers that equation simply isn't true for numbers.

http://www.space.com/scienceastronomy/time_theory_030806.html
quote:
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"There isn't a precise instant underlying an object's motion," he said. "And as its position is constantly changing over time -- and as such, never determined -- it also doesn't have a determined position at any time."
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Heisenberg uncertainty: DxDp >= hbar/2 As the observed expansion of the universe continues, and, as the mean temperature of the universe continues to approach absolute zero, could the universe transform into a condition analogous to a "Bose Einstein condensate"?
At the foundation of classical logic is the law of excluded middle,
A|~A|A V ~A
T_|F||__T
F_|T||__T
This forms an invariance principle which is a symmetry
T|F = F|T = T

Remarks added 22 May 2002: The remarks below are given as they were in a (memo) note that wasn't generally accessible. Now I am not really updating it, but since the equation (vacuum) itself is now included on my "web page" it is time also to include these remarks. At the present time I think the "input" of the gravitational action of matter, etc. might be studied in terms of boundary value problems. Then on one side of a boundary there could be the vacuum equation to be satisfied. And there are some ideas that relate to this. But these are ideas that call for further study.
3 memo of May 31, 2001: The equation is tensor equation which has a parallel or similarity to "wave" equations and can be described in terms of a d'Alembertian operator. It is thought of as of interest as an alternative description of the general relativistic space-time continuum that allows for "compressional" waves rather than allowing only for "transverse" waves. At the present time I am still seeking to find a good "input relation" for matter as the source of gravitation (analogous to the relation found by Einstein and Hilbert for the 2nd order tensor equation of standard GR). The vacuum equation can be described as having (on a LHS side that is equated to zero) a fourth order term formed by the covariant d'Alembertian operating on the G-tensor of Einstein plus an additive portion of second order (as to the differentiation) formed by quadratic combinations of curvature tensor elements. The precise additive portion or set of terms is defined by the condition that the total LHS is so structured so as to be formally divergence free (like the G-tensor is intrinsically divergence free). The plan is to put into this directory, ultimately, files of graphic type including the tensor equations in handwritten form.

Completed infinities, called "alephs" are distributive in nature, similar to the way that a set of "red" objects has the distributive property of redness[qualia]. Properties, or "attributes" like red are numbers in the sense that they interact algebraically according to the laws of Boolean algebra. Take one object away from the set of red objects and the distributive number "red" still describes the set. The distributive identity[attribute] "natural number" or "real number" describes an entire collection of individual objects.

The alephs can be set into a one to one correspondence with a proper subset of of themselves. The "infinite" Cantorian alephs are really distributive[qualia].

Yet, if we have a finite set of 7 objects, the cardinal number 7 does not really distribute over its individual subsets. Take anything away from the set and the number 7 ceases to describe it[wave function collapse-condensation into specific localization].

Symmetry is analogous to a generalized form of self evident truth, and it is a distributive attribute via the laws of nature, being distributed over the entire system called universe. A stratification of Cantorian alephs with varying degrees of complexity. Less complexity = greater symmetry = higher infinity-alephs. So the highest aleph, the "absolute-infinity" distributes over the entire set called Universe and gives it "identity".

The highest symmetry is a distributive mathematical identity[also a total unknown but possibly analogous to a state of "nothingness"]. This fact is reflected in part, by the conservation laws.

So an unbound-infinite-potentia and a constrained-finite-bound-actuality, are somehow different yet the same. The difference and sameness relation is a duality. Freedom(higher symmetry) and constraint-complexity-organizational structure(lesser symmetry) form a relation that can be described by an invariance principle.

On a flat Euclidean surface, the three angles of a triangle sum to 180 degrees. On the curved surface of a sphere, the three angles add up to more than 180 degrees. On the hyperbolic surface of a saddle they sum to less than 180 degrees. The three types of surface are not equivalent.

There is a rotational invariance for a triangle, that seems to hold for the three types of surface though.

An abstract representation is exactly that, "abstract". It is not a space, or time, but is instead a product of consciousness, or a mental construct. Topologically it is equivalent to a "point". The abstract description contains the concrete topology. Likewise, the concrete contains the abstract.

A duality.

A point contains an infinite expanse of space and time?

Could it be, that the "absolute" infinity, is actually a dimensionless point? Or more correctly, an "infinitesimal"?

Universe? = Zero?

On one level of stratification, two photons are separate. On another level, of stratification, the photons have zero separation.

Instantaneous communication between two objects, separated by a distance interval, is equivalent to zero separation[zero boundary] between the two objects.

According to the book "Gravitation", chapter 15, geometry of spacetime gives instructions to matter telling matter to follow the straightest path, which is a geodesic. Matter in turn, tells spacetime geometry how to curve in such a way, as to guarantee the conservation of momentum and energy. The Einstein tensor[geometric feature-description] is also conserved in this relationship between matter and the spacetime geometry. Eli Cartan's "boundary of a boundary equals zero."

A point can be defined as an "infinitesimal". The Topological spaces are defined as being diffeomorphism invariant. Intersecting cotangent bundles[manifolds] are the set of all possible configurations of a system, i.e. they describe the phase space of the system.

Waves are then abstract distributions and particles are convergent "concrete" localizations.

Quantum mechanics leads us to the realization that all matter-energy can be explained in terms of "waves". In a confined region(i.e. a closed universe or a black hole) the waves exists as STANDING WAVES In a closed system, the entropy never decreases.

The analogy with black holes is an interesting one but if there is nothing outside the universe, then it cannot be radiating energy outside itself as black holes are explained to be. So the amount of information i.e. "quantum states" in the universe is increasing. We see it as entropy, but to an information processor with huge computational capabilities, it is compressible information.

Quantum field theory calculations where imaginary time is periodic, with period 1/T are equivalent to statistical mechanics calculations where the temperature is T. The periodic waveforms that are opposed yet "in phase" would be at standing wave resonance, giving the action.

Periodicity is a symmetry. Rotate into the complex plane and we have
real numbers on the horizonal axis and imaginary numbers on the
vertical axis. So a periodic function could exist with periodicity
along both the imaginary AND the real axis. Such functions would have
amazing symmetries. Functions that remain unchanged, when the complex
variable "z" is changed.

f(z)---->f(az+b/cz+d)

Where the elements a,b,c,d, are arranged as a matrix, forming an
algebraic group. An infinite number of possible variations that
commute with each other as the function f, is invariant under group
transformations. These functions are known as "automorphic forms".

Topologically speaking, the wormhole transformations must be
invariant with regards to time travel. In other words, by traveling
backwards in time, we "complete" the future, and no paradoxes are
created.

So when spacetime tears and a wormhole is created, it must obey
certain transformative rules, which probably appear to be
discontinuities from a "3-D" perspective, but really, these
transformations are continuous!

So the number of holes[genus] on the surface of space, determine
whether there exist an infinite, or finite, number of solutions to
the universal equations?

The metric space has distance function r(x,y), definitively characterized by involvement with the real numbers, R, such that the metric space and R are embedded simultaneously in the full structure of manifold M. A topological space consists of sets of points which are defined[in this case] to be the intersections of cotangent bundles.

We move on to functions of a complex variable in the 2-D Euclidean space Z, utilizing the algebraic structure of complex numbers, where points of Z may be regarded as pairs of real numbers, R. Formal statements concerning Z, are also expressable as statements about R.

Complex-valued functions e.g. w = f(z), can also be represented as binary, or quaternary relations, these sets of complex valued polynomials then have elements that map Z into Z. Every complex valued polynomial subset p(z), has natural number coefficients a_k, or the "kth" coefficient of p(z)..

So, for non-zero polynomials, we take into account the number of zeros OF the polynomial O[p(z)] with O ranging over the natural numbers such that
O[p(z), R ] = v, where v is a finite natural number.

Thus if G is a metric group and E is a topological group, such, that an open neighborhood U, of the identity is a metric space, with a compliant metric, specifiable to the topology of U, then the distance r(a,b) between any two points (a,b) in U, then in terms of the distance u(E) = J, consists of the points a of U with the fact that r(a,E) is an infinitesimal.

"The metric space has distance function r(x,y), definitively characterized by involvement with the real numbers, R, such that the metric space and R are embedded simultaneously in the full structure of manifold M. A topological space consists of sets of points which are defined[in this case] to be the intersections of cotangent bundles."

Would you care to elaborate? Let X be some abstract space (possibly even containing a proper subclass) with the trivial metric (r(x,y) =0 if x=y 1 otherwise.) How does one embed this in a manifold? which manifold, what do you mean by embed? I mean as X is not a manifold, you don't mean the usual embedding (or immersion or submersions) of differential geometry. As the metric only takes exactly two values, how do you relate it to all of R? Must every manifold contain an *embedded* copy of R? I can't remember the explicit definititions for embeddings, you see - I was never very good at memorizing all those things, you know, an immersion that's not this is not the other.

If f(x) is a homeomorphism from T onto S, and for every point p in T, f(U(p)) = U(f(p)), and the monad is invariant under standard topological transformations, with the caveat that the definition also comprizes a type of dynamic situation sematics, where concepts, such as "proper set", "ordinal" and "cardinal" are relativised to context, taking care of paradox at all levels via symmetry, or an invariant many-valued logic, and the "top[set of all sets]", would naturally not exist, of course, since there is nothing outside the universe. So it becomes an infinite chain or composition of ever more inclusive situated sets expressing an interesting informational -topological dynamic.

Outside of "Total Existence"[TE] there is nothing. This is an irrefutable fact. Or we could say that there IS not an "outside" of Total Existence.

Therefore the largest possible set does not exist, where "does not exist" is equivalent to "nothing".

If space is *quantized* yet also continuous, then it too, has the property called "wave-particle" duality. If space consists of indivisible units, then a measurement of space means that Fermat's last theorem holds, for it.

To every set A and every condition S(x) there corresponds a set B whose elements are exactly those elements x of A for which S(x) holds. This is the axiom that leads to Russell's paradox. For if we let the condition S(x) be: not(x element of x), then B = {x in A such that x is not in x}. Is B a member of B? If it is, then it isn't; and if it isn't, then it is. Therefore B cannot be in A, meaning that nothing contains everything.

This means that relativity holds in the "topological" sense and T-duality is correct.

Quantum entities are described as probability distributions, which are attributes of an underlying phase space, where the properties-attributes such as "spin" and "charge" are not the attributes of individual particles, but they are universally distributive entities, being the attributes of a "coherent wave function". It is this wave-distribution property that then "decoheres" into the ostensible "wave function collapse", as waves become localized particles that are "in phase" creating standing-spherical-wave resonances, which are condensations of space itself. The continual collapse-condensation of space into matter-energy is the continual "change", i.e. the property called "time". The spherical waves, or probability distributions are represented by the Schrodinger wave function, "psi".

The information density of the universal system must be increasing. The increase of information density is analogous to a pressure gradient.

[density 1]--->[density 2]--->[density 3]---> ... --->[density n]

[<-[->[<-[-><-]->]<-]->]

Intersecting wavefronts = increasing density of spacelike slices

As the wavefronts intersect, it becomes a mathematical computation:

2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...2^n

According to conventional theories, the surface area of the horizon surrounding a black hole, measures its entropy, where entropy is defined as a measure of the number of internal states that the black hole can be in without looking different to an outside observer, who must measure only mass, rotation, and charge. Another theory states that the maximum entropy of any closed region of space can never exceed one quarter of the area of the circumscribing surface, with the entropy being the measure of the total information contained by the system.

S' = S_m + A/4

So the "black hole" theorists came to realize that the information associated with all phenomena in the three dimensional world, can be stored on a two dimensional boundary, analogous to the storing of a holographic image.

Since entropy can also be defined as the number of states, that particles can be in within within a region of space, and the entropy of the universe must always increase, the next logical step is to realize that the spacetime density, i.e. the information encoded within a circumscribed region of space, must be increasing in the thermodynamic direction of time.

Of course, thermodynamic entropy is popularly described as the disorder or "randomness" in a physical system. In 1877, the physicist Ludwig Boltzmann defined entropy more precisely. He defined it in terms of the number of distinct microscopic states that the particles in a system can be configured, while still looking like the same macroscopic system. For example, a system such as a gas cloud, one would count the ways that the individual gas molecules could be distributed, and moving.

In1948, mathematician Claude E. Shannon, introduced today's most widely used measure of information content: entropy. The Shannon entropy of a message is the number of binary digits, i.e. "bits" needed to encode it. While the structure, quality, or value, of the information in Shannon entropy may be an unknown, the quantity of information can be known. Shannon entropy and thermodynamic entropy are equivalent.

The universal laws of nature are explained in terms of symmetry. The completed infinities, mathematician Georg Cantor's infinite sets, could be explained as cardinal identities, akin to "qualia" [Universally distributed attributes] from which finite subsets, and elements of subsets [quantum decoherence-wave function collapse] can be derived.

Completed infinities, called "alephs" are distributive in nature, similar to the way that a set of "red" objects has the distributive property of redness[qualia]. Properties, or "attributes" like red are numbers in the sense that they interact algebraically according to the laws of Boolean algebra. Take one object away from the set of red objects and the distributive number "red" still describes the set. The distributive identity[attribute] "natural number" or "real number" describes an entire collection of individual objects.

The alephs can be set into a one to one correspondence with a proper subset of of themselves. The "infinite" Cantorian alephs are really distributive[qualia].

Yet, if we have a finite set of 7 objects, the cardinal number 7 does not really distribute over its individual subsets. Take anything away from the set and the number 7 ceases to describe it[wave function collapse-condensation into specific localization].

Symmetry is analogous to a generalized form of self evident truth, and it is a distributive attribute via the laws of nature, being distributed over the entire system called universe. A stratification of Cantorian alephs with varying degrees of complexity. Less complexity = greater symmetry = higher infinity-alephs. So the highest aleph, the "absolute-infinity" distributes over the entire set called Universe and gives it "identity".

The highest symmetry is a distributive mathematical identity[also a total unknown but possibly analogous to a state of "nothingness"]. This fact is reflected in part, by the conservation laws.

So an unbound-infinite-potentia and a constrained-finite-bound-actuality, are somehow different yet the same. The difference and sameness relation is a duality. Freedom(higher symmetry) and constraint-complexity-organizational structure(lesser symmetry) form a relation that can be described by an invariance principle.

On a flat Euclidean surface, the three angles of a triangle sum to 180 degrees. On the curved surface of a sphere, the three angles add up to more than 180 degrees. On the hyperbolic surface of a saddle they sum to less than 180 degrees. The three types of surface are not equivalent.

There is a rotational invariance for a triangle, that seems to hold for the three types of surface though.

ABC = BCA = CAB

CBA = BAC = ACB

According to Einstein, and the CTMU of Langan, www.ctmu.org , "space and time are modes by which we think, and not conditions in which we live". Space becomes abstract, a relation that is perceptual and "mental", where distance interval between two points becomes a mental perception.

An abstract representation is exactly that, "abstract". It is not a space, or time, but is instead a product of consciousness, or a mental construct. Topologically it is equivalent to a "point". The abstract description contains the concrete topology. Likewise, the concrete contains the abstract.

A duality.

A point contains an infinite expanse of space and time?

Could it be, that the "absolute" infinity, is actually a dimensionless point? Or more correctly, an "infinitesimal"?

Universe? = Zero?

On one level of stratification, two photons are separate. On another level, of stratification, the photons have zero separation.

Instantaneous communication between two objects, separated by a distance interval, is equivalent to zero separation[zero boundary] between the two objects.

According to the book "Gravitation", chapter 15, geometry of spacetime gives instructions to matter telling matter to follow the straightest path, which is a geodesic. Matter in turn, tells spacetime geometry how to curve in such a way, as to guarantee the conservation of momentum and energy. The Einstein tensor[geometric feature-description] is also conserved in this relationship between matter and the spacetime geometry. Eli Cartan's "boundary of a boundary equals zero."

A point can be defined as an "infinitesimal". The Topological spaces are defined as being diffeomorphism invariant. Intersecting cotangent bundles[manifolds] are the set of all possible configurations of a system, i.e. they describe the phase space of the system.